Question

1. after kicking, one coach likes to play practice games with 6 players on each team. she doesn’t like any team to have more or fewer than 6 players and she doesn’t like anyone to be left off a team. what are the possible numbers of players who must show up to practice so that the coach can make these teams?

Explanation:

Step1: Understand the problem

The coach wants teams with 6 players each, and no one left out. So we need to find numbers of players that are multiples of 6 (since each team has 6, and no remainder). Also, the number can't be more or less than 6 per team, so we consider non - negative integer multiples of 6. Let the number of teams be \(n\) (where \(n\) is a positive integer, \(n = 1,2,3,\cdots\)) and the number of players be \(P\). The relationship is \(P=6\times n\).

Step2: Find possible values

For \(n = 1\), \(P = 6\times1=6\)
For \(n = 2\), \(P = 6\times2 = 12\)
For \(n = 3\), \(P=6\times3 = 18\)
And so on. In general, the number of players \(P\) must be a positive multiple of 6, i.e., \(P = 6k\), where \(k\in\mathbb{Z}^+\) (positive integers). But if we consider practical cases (like a reasonable number of teams), common possible numbers start from 6, 12, 18, 24, etc. But since the problem doesn't specify a maximum number of teams, the set of possible numbers of players is all positive integer multiples of 6. However, if we assume a small number of teams (like at least 1 team and maybe up to a reasonable limit, but the problem doesn't specify, so we can say the possible numbers are 6, 12, 18, ... (multiples of 6)). But if we consider the context of a practice game, maybe the minimum is 6 (1 team) and then 12 (2 teams), 18 (3 teams) etc.

Answer:

The possible number of players are positive integer multiples of 6, such as 6, 12, 18, 24, etc. If we consider the smallest non - zero number of teams (at least 1 team), the possible numbers start from 6, 12, 18, and so on.